If you wanted to go by spell level, along the same vein as the darkness/daylight spells, then gust of wind wouldn't be able to put out a wall of flame unless it is cast as a fourth level spell.
From page 227, PHB:
The gust disperses gas or vapor, and it extinguishes candles, torches, and similar unprotected flames in the area. It causes protected flames, such as those of lanterns, to dance wildly and has a 50 percent chance to extinguish them.
Using the listed examples as the rule, Wall of Fire would be immune to being extinguished by Gust of Wind, as it is a much larger conflagration than a mere candle or torch.
From page 263, PHB:
You can make the wall up to 60 feet long, 20 feet high, and 1 foot thick, or a ringed wall up to 20 feet in diameter, 20 feet high, and 1 foot thick. The wall is opaque and lasts for the duration.
A flame wall that is dense enough to be opaque is large and strong enough that a gust of wind won't be able to put it out. Imagine a raging inferno the size of a house, and imagine trying to put it out with air. You'd need hurricane force winds to extinguish the flames, especially since pouring air onto a fire just feeds it oxygen faster.
I don't think the source of the flame matters as much as its size. Produce flame would work well for lighting a torch, or setting easily combustible material alight (like a bale of hay), but wouldn't really stand up against a hurricane. Any fire can be extinguished when the combustion material is consumed or its access to oxygen is blocked. Wind is great for putting out smaller flames, but would do poorly against larger ones because it can't effectively smother the entire fire at once (a very poor model for describing the dynamics of fire, but it is approximately accurate). Water would be a much better choice for taking out a fire, magical or not because it can smother the entire flame, blocking the flame from accessing the combustion material (probably wood or coal) and blocking access to oxygen.